1 00:00:24,970 --> 00:00:29,150 PROFESSOR: Hi, I'm Gilbert Strang, and this is the very 2 00:00:29,150 --> 00:00:35,870 first in a series of videos about highlights of calculus. 3 00:00:35,870 --> 00:00:40,200 I'm doing these just because I hope they'll be helpful. 4 00:00:40,200 --> 00:00:47,790 It seems to me so easy to be lost in the big calculus 5 00:00:47,790 --> 00:00:53,300 textbooks and the many, many problems and in the details. 6 00:00:53,300 --> 00:00:56,660 But do you see the big picture? 7 00:00:56,660 --> 00:01:00,370 Well, I hope this will help. 8 00:01:00,370 --> 00:01:04,150 For me, calculus is about the 9 00:01:04,150 --> 00:01:09,110 relation between two functions. 10 00:01:09,110 --> 00:01:13,990 And one example for those two functions, one good example, 11 00:01:13,990 --> 00:01:20,760 is function 1, the distance, distance traveled, what you 12 00:01:20,760 --> 00:01:23,890 see on a trip meter in a car. 13 00:01:23,890 --> 00:01:30,790 And function 2, the one that goes with distance, is speed, 14 00:01:30,790 --> 00:01:34,720 how quickly you're going, how fast you're traveling. 15 00:01:34,720 --> 00:01:37,330 So that's one pair of functions. 16 00:01:37,330 --> 00:01:40,570 Let me give another pair. 17 00:01:40,570 --> 00:01:44,450 I could get more and more, but I think if we get these two 18 00:01:44,450 --> 00:01:49,190 pairs, we can move forward. 19 00:01:49,190 --> 00:01:54,610 So in this second pair, height is function 1, 20 00:01:54,610 --> 00:01:57,860 how high you've climbed. 21 00:01:57,860 --> 00:02:03,540 If it's a graph, how far the graph goes above the axis. 22 00:02:03,540 --> 00:02:06,060 Up, in other words. 23 00:02:06,060 --> 00:02:11,600 So that's height, and then the other one tells you 24 00:02:11,600 --> 00:02:13,460 how fast you climb. 25 00:02:13,460 --> 00:02:15,730 The height tells how far you climbed. 26 00:02:15,730 --> 00:02:18,070 It could be a mountain. 27 00:02:18,070 --> 00:02:22,300 And then the slope tells you how quickly you're climbing at 28 00:02:22,300 --> 00:02:23,050 each point. 29 00:02:23,050 --> 00:02:25,630 Are you going nearly straight up? 30 00:02:25,630 --> 00:02:26,620 Flat? 31 00:02:26,620 --> 00:02:28,790 Possibly down? 32 00:02:28,790 --> 00:02:34,400 So distance and speed, height and slope will serve as good 33 00:02:34,400 --> 00:02:36,660 examples to start with. 34 00:02:36,660 --> 00:02:42,480 And let me give you some letters, some algebra letters 35 00:02:42,480 --> 00:02:45,090 that you might use. 36 00:02:45,090 --> 00:02:53,490 Distance, maybe I would call that f of t. 37 00:02:53,490 --> 00:03:01,480 So f for how far or for function, and the idea is that 38 00:03:01,480 --> 00:03:04,990 t is the input. 39 00:03:04,990 --> 00:03:07,970 It's the time when you're asking for the distance. 40 00:03:07,970 --> 00:03:10,060 The output is the distance. 41 00:03:10,060 --> 00:03:16,580 Or in the case of height, maybe y of x would 42 00:03:16,580 --> 00:03:18,740 be the right one. 43 00:03:18,740 --> 00:03:22,270 x is how far you go across. 44 00:03:22,270 --> 00:03:23,870 That's the input. 45 00:03:23,870 --> 00:03:28,830 And at each x, you have an output y how far up? 46 00:03:28,830 --> 00:03:31,860 So f is telling you how far. 47 00:03:31,860 --> 00:03:36,660 y is telling you the height of a graph. 48 00:03:36,660 --> 00:03:41,340 That's function 1, two examples of function 1. 49 00:03:41,340 --> 00:03:43,060 Now, what about slope? 50 00:03:43,060 --> 00:03:46,560 Well, luckily, speed and slope start with the same letter, so 51 00:03:46,560 --> 00:03:52,990 I'll often use s for the speed or the slope for this second-- 52 00:03:52,990 --> 00:03:56,960 oh, it even stands for a second function. 53 00:03:56,960 --> 00:04:00,590 But let me tell you also the right-- 54 00:04:00,590 --> 00:04:01,620 the official-- 55 00:04:01,620 --> 00:04:06,670 letters that make the connection between function 2 56 00:04:06,670 --> 00:04:09,030 and function 1. 57 00:04:09,030 --> 00:04:15,240 If my function is a function of time, the distance, how far 58 00:04:15,240 --> 00:04:19,589 I go, then the speed is-- 59 00:04:19,589 --> 00:04:27,050 the right letters are df dt. 60 00:04:27,050 --> 00:04:29,840 Everybody uses those letters. 61 00:04:29,840 --> 00:04:32,830 So let me say again how to pronounce: df dt. 62 00:04:35,710 --> 00:04:40,100 And Leibniz came up with that notation, and 63 00:04:40,100 --> 00:04:42,850 he just got it right. 64 00:04:42,850 --> 00:04:44,920 And what would this one be? 65 00:04:44,920 --> 00:04:56,760 Well, corresponding to this, it looks the same, or dy dx. 66 00:04:56,760 --> 00:05:01,060 Again, I'll just repeat how to say that: dy dx. 67 00:05:01,060 --> 00:05:06,940 And that is the slope, and we have to understand what those 68 00:05:06,940 --> 00:05:07,820 symbols mean. 69 00:05:07,820 --> 00:05:09,730 Right now, I'm just writing them down as symbols. 70 00:05:13,070 --> 00:05:15,740 May I begin with the most important and the simplest 71 00:05:15,740 --> 00:05:17,650 example of all? 72 00:05:17,650 --> 00:05:19,660 Let me take that case. 73 00:05:19,660 --> 00:05:25,590 OK, so the key example here, the one to get completely 74 00:05:25,590 --> 00:05:29,450 straight is the case of constant 75 00:05:29,450 --> 00:05:32,820 speed, constant slope. 76 00:05:32,820 --> 00:05:34,900 I'll just graph that. 77 00:05:34,900 --> 00:05:36,225 So here I'm go to graph. 78 00:05:39,890 --> 00:05:43,220 Shall I make it the speed? 79 00:05:43,220 --> 00:05:45,000 Yeah, let's say speed. 80 00:05:45,000 --> 00:05:48,600 So time is going along that way. 81 00:05:48,600 --> 00:05:51,410 Speed is up this way. 82 00:05:51,410 --> 00:05:56,740 And I'm going to say in this first example that the speed 83 00:05:56,740 --> 00:05:57,690 is the same. 84 00:05:57,690 --> 00:06:00,250 We're traveling at the constant speed 85 00:06:00,250 --> 00:06:03,610 of let's say 40. 86 00:06:03,610 --> 00:06:06,920 So it stays at the height of 40. 87 00:06:06,920 --> 00:06:13,040 Oh, properly, I should add units like miles per hour or 88 00:06:13,040 --> 00:06:17,520 kilometers per hour or meters per second or whatever. 89 00:06:21,600 --> 00:06:24,680 For now, I'll just write 40. 90 00:06:24,680 --> 00:06:32,330 OK, now if we're traveling at a speed of 40 miles per hour, 91 00:06:32,330 --> 00:06:34,780 what's the distance? 92 00:06:34,780 --> 00:06:42,300 Well, let me start with the trip meter at zero. 93 00:06:42,300 --> 00:06:45,790 so this is time again, and now this is 94 00:06:45,790 --> 00:06:47,040 going to be the distance. 95 00:06:49,820 --> 00:06:55,000 After one hour, my distance is 40. 96 00:06:55,000 --> 00:06:59,990 So if I mark t equal to 1, I've reached 40. 97 00:06:59,990 --> 00:07:01,940 That's height of 40. 98 00:07:01,940 --> 00:07:06,175 At t equal to 2, I've reached 80. 99 00:07:06,175 --> 00:07:10,420 At t equal to 1/2, half an hour, I've reached 20. 100 00:07:10,420 --> 00:07:14,830 Those points lie on a line. 101 00:07:14,830 --> 00:07:21,120 The graph of distance covered when you're just traveling at 102 00:07:21,120 --> 00:07:27,890 a steady rate, constant rate, constant speed is just a 103 00:07:27,890 --> 00:07:30,070 straight line. 104 00:07:30,070 --> 00:07:32,660 And now I can make the connection. 105 00:07:32,660 --> 00:07:37,500 I've been speaking here about distance and speed. 106 00:07:37,500 --> 00:07:40,170 But now let me think of this as the height-- 107 00:07:40,170 --> 00:07:41,450 40 is that height. 108 00:07:41,450 --> 00:07:45,090 80 is that height-- 109 00:07:45,090 --> 00:07:46,740 and ask about slope. 110 00:07:46,740 --> 00:07:48,180 What is slope? 111 00:07:48,180 --> 00:07:52,810 So let's just remember what's the connection here. 112 00:07:52,810 --> 00:07:57,720 What's the slope if that's the distance if I look at my trip 113 00:07:57,720 --> 00:08:02,970 meter and I know I'm traveling along at that constant speed, 114 00:08:02,970 --> 00:08:05,700 how do I find that speed? 115 00:08:05,700 --> 00:08:13,670 Well, slope, it's the distance up, which would be 40 after 116 00:08:13,670 --> 00:08:15,980 one hour, divided by the distance 117 00:08:15,980 --> 00:08:23,820 across, 40/1, or 80/2. 118 00:08:23,820 --> 00:08:26,760 Doesn't matter, because we're traveling at constant speed, 119 00:08:26,760 --> 00:08:39,110 so the slope, which is up, over, across is 40/1, 120 00:08:39,110 --> 00:08:45,570 80/2, 20 over 1/2. 121 00:08:45,570 --> 00:08:51,360 I'll put 80/2 as one example: 40. 122 00:08:51,360 --> 00:08:55,780 Oh, let me do it-- 123 00:08:55,780 --> 00:08:56,960 that's arithmetic. 124 00:08:56,960 --> 00:08:59,240 Let me do it with algebra. 125 00:08:59,240 --> 00:09:02,000 We don't need calculus yet, by the way. 126 00:09:02,000 --> 00:09:05,060 Calculus is coming pretty quickly. 127 00:09:05,060 --> 00:09:08,090 This is the step we can take. 128 00:09:08,090 --> 00:09:15,370 Because the speed is constant, we can just divide the 129 00:09:15,370 --> 00:09:22,760 distance by the time to find-- 130 00:09:22,760 --> 00:09:26,020 and this slope, let me right speed also. 131 00:09:28,870 --> 00:09:33,620 Up, over, across, distance over time, f/t, 132 00:09:33,620 --> 00:09:35,280 that gives us s. 133 00:09:35,280 --> 00:09:36,530 This is s. 134 00:09:39,500 --> 00:09:43,590 OK, what about-- 135 00:09:43,590 --> 00:09:46,400 calculus goes both ways. 136 00:09:46,400 --> 00:09:48,210 We can go both ways here. 137 00:09:48,210 --> 00:09:50,500 We already have practically. 138 00:09:50,500 --> 00:09:56,140 Here I went in the direction from 1 to 2. 139 00:09:56,140 --> 00:09:58,090 Now, I want to go in the direction-- 140 00:09:58,090 --> 00:10:00,090 suppose I know the speed. 141 00:10:00,090 --> 00:10:03,490 How do I recover the distance? 142 00:10:03,490 --> 00:10:08,560 If I know my speed is 40 and I know I started at zero, what's 143 00:10:08,560 --> 00:10:09,810 my distance? 144 00:10:11,780 --> 00:10:14,740 Distance or height, either one. 145 00:10:19,970 --> 00:10:24,830 So these are like both. 146 00:10:24,830 --> 00:10:26,280 Now, I'm just going the other way. 147 00:10:26,280 --> 00:10:27,670 Well, you see how. 148 00:10:27,670 --> 00:10:29,060 How do I find f? 149 00:10:32,294 --> 00:10:36,630 It's s times t, right? 150 00:10:36,630 --> 00:10:39,560 Your algebra automatically says if you see a t there, you 151 00:10:39,560 --> 00:10:41,490 can put it there. 152 00:10:41,490 --> 00:10:43,250 So it's s times t. 153 00:10:43,250 --> 00:10:45,640 It's a straight line. 154 00:10:45,640 --> 00:10:50,150 s times t, s times x, y equal sx. 155 00:10:50,150 --> 00:10:51,400 Let me put another-- 156 00:10:55,050 --> 00:10:59,310 the same idea with my y, x letters. 157 00:10:59,310 --> 00:11:00,610 It's that line. 158 00:11:00,610 --> 00:11:05,030 In other words, if that one is constant, this one is a 159 00:11:05,030 --> 00:11:07,280 straight line. 160 00:11:07,280 --> 00:11:12,400 OK, straightforward, but very, very fundamental. 161 00:11:12,400 --> 00:11:15,290 In fact, can I call your attention to something a 162 00:11:15,290 --> 00:11:17,780 little more? 163 00:11:17,780 --> 00:11:27,280 Suppose I measured between time 2 and time 1. 164 00:11:27,280 --> 00:11:31,330 So I'm looking between time 2 and time 1, and I look how far 165 00:11:31,330 --> 00:11:32,860 I went in that time. 166 00:11:36,760 --> 00:11:39,150 But what I'm trying is-- 167 00:11:39,150 --> 00:11:41,710 I'm going to put in another little symbol because it's 168 00:11:41,710 --> 00:11:44,720 going to be really worth knowing. 169 00:11:44,720 --> 00:11:52,430 It's really the change in f divided by the change in t. 170 00:11:52,430 --> 00:11:58,210 I use that letter delta to indicate 171 00:11:58,210 --> 00:12:00,800 a difference between-- 172 00:12:00,800 --> 00:12:05,110 the difference between time 2 and time 1 was 1, and the 173 00:12:05,110 --> 00:12:11,130 difference between height 2 and height 1 was 40. 174 00:12:11,130 --> 00:12:14,390 You see, I'm looking at this little piece. 175 00:12:14,390 --> 00:12:17,710 And, of course, the slope is still 40. 176 00:12:17,710 --> 00:12:21,350 It's still the slope of that line. 177 00:12:21,350 --> 00:12:26,800 Yeah, so that really what I'm measuring in speed there, I 178 00:12:26,800 --> 00:12:30,670 don't always have to be starting at t equals 0, and I 179 00:12:30,670 --> 00:12:33,390 don't always have to be starting at f equals 0. 180 00:12:33,390 --> 00:12:34,880 Oh, let me draw that. 181 00:12:34,880 --> 00:12:38,100 Suppose I started at f equals 40. 182 00:12:38,100 --> 00:12:40,720 My trip meter happened to start at 40. 183 00:12:40,720 --> 00:12:43,300 After an hour, I'd be up to 80. 184 00:12:43,300 --> 00:12:46,270 After another hour, I'd be up to 120. 185 00:12:46,270 --> 00:12:50,430 Do you see that this starting the trip meter, who cares 186 00:12:50,430 --> 00:12:53,160 where the trip meter started? 187 00:12:53,160 --> 00:12:55,450 It's the change in the trip meter that tells how 188 00:12:55,450 --> 00:12:57,520 long the trip was. 189 00:12:57,520 --> 00:12:58,560 Clear. 190 00:12:58,560 --> 00:13:01,383 OK, so that's that example. 191 00:13:06,450 --> 00:13:14,020 We come back to it because it's the basic one where the 192 00:13:14,020 --> 00:13:15,950 speed is constant. 193 00:13:15,950 --> 00:13:20,170 And even if now I have to move to a changing speed, you have 194 00:13:20,170 --> 00:13:24,210 to let me bring calculus into these lectures. 195 00:13:24,210 --> 00:13:28,290 OK, I'm going to draw another picture, and you 196 00:13:28,290 --> 00:13:32,692 tell me about the-- 197 00:13:32,692 --> 00:13:38,310 yeah, let me draw function 1, another example of function 1. 198 00:13:38,310 --> 00:13:40,600 So again I have time. 199 00:13:40,600 --> 00:13:42,280 I have distance. 200 00:13:42,280 --> 00:13:47,280 I'm going to start at zero, but I'm not going to keep the 201 00:13:47,280 --> 00:13:48,610 speed constant. 202 00:13:48,610 --> 00:13:53,520 I'm going to start out at a good speed, but I'm 203 00:13:53,520 --> 00:13:55,805 going to slow down. 204 00:13:55,805 --> 00:13:59,590 Do you see me slowing down there? 205 00:13:59,590 --> 00:14:01,340 I don't mean slowing down with the chalk. 206 00:14:01,340 --> 00:14:05,650 I mean slowing down with slope. 207 00:14:05,650 --> 00:14:08,690 The slope started out steep. 208 00:14:08,690 --> 00:14:12,800 By here, by that point, the slope was zero. 209 00:14:12,800 --> 00:14:16,050 What was the car doing here? 210 00:14:16,050 --> 00:14:21,860 The car is certainly moving forward because the distance 211 00:14:21,860 --> 00:14:23,970 is increasing. 212 00:14:23,970 --> 00:14:26,770 Here it's increasing faster. 213 00:14:26,770 --> 00:14:30,480 Here it's increasing barely. 214 00:14:30,480 --> 00:14:33,810 In other words, we're putting on the brakes. 215 00:14:33,810 --> 00:14:35,130 The car is slowing down. 216 00:14:35,130 --> 00:14:36,540 We're coming to a red light. 217 00:14:36,540 --> 00:14:40,860 In fact, there is the red light right at that time. 218 00:14:40,860 --> 00:14:48,030 Now, just stay with it to think what would the speed 219 00:14:48,030 --> 00:14:49,840 look like for this problem? 220 00:14:53,430 --> 00:14:55,850 If that's a picture of the function, just 221 00:14:55,850 --> 00:14:57,410 let's get some idea. 222 00:14:57,410 --> 00:15:03,210 I'm not going to have a formula yet. 223 00:15:03,210 --> 00:15:05,170 I'm not putting in all the details. 224 00:15:05,170 --> 00:15:11,230 Well, actually, I don't plan to put in all the details of 225 00:15:11,230 --> 00:15:18,070 calculus of every possible step we might take. 226 00:15:18,070 --> 00:15:21,070 It's the important ones I'm hoping to show you and I'm 227 00:15:21,070 --> 00:15:24,890 hoping for you to see that they are important. 228 00:15:24,890 --> 00:15:27,430 OK, what is important? 229 00:15:27,430 --> 00:15:29,820 Roughly, what does the graph looks like? 230 00:15:29,820 --> 00:15:33,060 Well, the speed-- 231 00:15:33,060 --> 00:15:34,100 the slope-- 232 00:15:34,100 --> 00:15:37,800 started out somewhere up there. 233 00:15:37,800 --> 00:15:45,040 Yeah, it started out at a good speed and slowed down. 234 00:15:45,040 --> 00:15:48,030 And by this point, ha! 235 00:15:48,030 --> 00:15:51,040 Let's mark that time here on that graph. 236 00:15:51,040 --> 00:15:55,930 Do you see what is the speed at that moment? 237 00:15:55,930 --> 00:16:00,770 The speed at that moment is zero. 238 00:16:00,770 --> 00:16:02,420 The car has stopped. 239 00:16:02,420 --> 00:16:04,500 The speed is decreasing. 240 00:16:04,500 --> 00:16:08,280 Let me make it decrease, decrease, decrease, decrease, 241 00:16:08,280 --> 00:16:13,860 and at that moment, the speed is zero right there. 242 00:16:13,860 --> 00:16:15,680 That's that point. 243 00:16:15,680 --> 00:16:21,400 See, two different pictures, two different functions. 244 00:16:21,400 --> 00:16:24,730 but same information. 245 00:16:24,730 --> 00:16:31,190 So calculus has the job of given one of those functions, 246 00:16:31,190 --> 00:16:32,300 find the other one. 247 00:16:32,300 --> 00:16:34,690 Given this function, find that one. 248 00:16:34,690 --> 00:16:36,380 This way is called-- 249 00:16:36,380 --> 00:16:41,040 from function one to function two, that's called 250 00:16:41,040 --> 00:16:43,690 differential calculus. 251 00:16:43,690 --> 00:16:46,230 Big, impressive word anyway. 252 00:16:46,230 --> 00:16:51,800 That's function one to two, finding the speed. 253 00:16:51,800 --> 00:16:57,860 Going the other direction is called integral calculus. 254 00:16:57,860 --> 00:17:03,500 The step is called integration when you take the speed over 255 00:17:03,500 --> 00:17:09,510 that period of time, and you recover the distance. 256 00:17:09,510 --> 00:17:12,569 So it's differential calculus in one direction, integral 257 00:17:12,569 --> 00:17:13,930 calculus in the other. 258 00:17:13,930 --> 00:17:15,880 Now, here's a question. 259 00:17:18,940 --> 00:17:22,230 Let me continue that curve a little longer. 260 00:17:22,230 --> 00:17:24,609 I got it to the red light. 261 00:17:24,609 --> 00:17:32,040 Now imagine that the distance starts going 262 00:17:32,040 --> 00:17:36,160 down from that point. 263 00:17:36,160 --> 00:17:39,080 What's happening? 264 00:17:39,080 --> 00:17:42,690 The distance is decreasing. 265 00:17:42,690 --> 00:17:45,360 The car is going backwards. 266 00:17:45,360 --> 00:17:48,020 It's going in reverse. 267 00:17:48,020 --> 00:17:51,700 The speed, what's the speed? 268 00:17:51,700 --> 00:17:53,860 Negative. 269 00:17:53,860 --> 00:18:01,430 The speed, because distance is going from higher to lower, 270 00:18:01,430 --> 00:18:03,320 that counts for negative speed. 271 00:18:03,320 --> 00:18:08,245 The speed curve would be going down here. 272 00:18:08,245 --> 00:18:13,730 Do you see that that's a not brilliantly drawn picture, but 273 00:18:13,730 --> 00:18:15,230 you're seeing the-- 274 00:18:15,230 --> 00:18:19,240 that's the farthest it went. 275 00:18:19,240 --> 00:18:24,660 Then the car started backwards, and the speed curve 276 00:18:24,660 --> 00:18:27,820 reflected that by going below zero. 277 00:18:27,820 --> 00:18:30,990 You see, two different curves, but same information. 278 00:18:35,950 --> 00:18:38,290 I'm remembering an old movie. 279 00:18:38,290 --> 00:18:41,250 I don't know if you saw an old B movie called Ferris 280 00:18:41,250 --> 00:18:43,920 Bueller's Day Off. 281 00:18:43,920 --> 00:18:45,440 Did you see that? 282 00:18:45,440 --> 00:18:49,080 So the kid had borrowed his father's-- 283 00:18:49,080 --> 00:18:55,570 not borrowed, but lifted his father's good car and drove it 284 00:18:55,570 --> 00:19:00,740 a lot like so and put on a lot of mileage. 285 00:19:00,740 --> 00:19:03,260 The trip meter was way up, and he knew his father was going 286 00:19:03,260 --> 00:19:05,390 to notice this. 287 00:19:05,390 --> 00:19:12,620 So he had the idea to put the car up on a lift, put it in 288 00:19:12,620 --> 00:19:16,820 reverse, and go for a while, and the trip 289 00:19:16,820 --> 00:19:18,160 meter would go backwards. 290 00:19:22,940 --> 00:19:24,755 I don't know if trip meters do go backwards. 291 00:19:29,380 --> 00:19:36,000 It's kind of tough to watch them while going in reverse. 292 00:19:36,000 --> 00:19:43,780 But if whoever made the car understood calculus, as you 293 00:19:43,780 --> 00:19:47,520 do, the speedometer-- 294 00:19:47,520 --> 00:19:49,680 now that I think of it, speedometers don't have a 295 00:19:49,680 --> 00:19:50,450 below zero. 296 00:19:50,450 --> 00:19:55,540 They should have. And trip meters should go backwards. 297 00:19:55,540 --> 00:19:59,690 I mean, that movie was just made for a calculus person. 298 00:20:05,830 --> 00:20:07,950 Maybe I'm remembering more. 299 00:20:07,950 --> 00:20:10,330 I think it didn't work or something. 300 00:20:10,330 --> 00:20:15,030 And the kid got mad and kicked the car, and it fell off the 301 00:20:15,030 --> 00:20:20,190 lift, went through the glass window. 302 00:20:20,190 --> 00:20:23,370 Anyway, calculus would have saved him if 303 00:20:23,370 --> 00:20:27,480 only the car had been-- 304 00:20:27,480 --> 00:20:31,500 or the meters in the car had been made correctly. 305 00:20:31,500 --> 00:20:36,320 All right, that's one pair. 306 00:20:36,320 --> 00:20:42,650 That's our first real pair in which the speed changes. 307 00:20:42,650 --> 00:20:44,580 OK. 308 00:20:44,580 --> 00:20:54,460 I thought in this first video, later, even today, I'll get to 309 00:20:54,460 --> 00:20:59,340 a case where we have formulas. 310 00:20:59,340 --> 00:21:03,200 That's what calculus moves into. 311 00:21:03,200 --> 00:21:08,420 When f of t is given by some formula, well, here it's given 312 00:21:08,420 --> 00:21:10,210 by a formula: s times t. 313 00:21:13,110 --> 00:21:15,020 A simple formula. 314 00:21:15,020 --> 00:21:20,350 And then, knowing that, we know that the speed is s. 315 00:21:20,350 --> 00:21:23,340 Later, we got more functions. 316 00:21:23,340 --> 00:21:29,030 But let me take an example, just because these pairs of 317 00:21:29,030 --> 00:21:32,030 functions are everywhere. 318 00:21:32,030 --> 00:21:33,970 What could I take? 319 00:21:33,970 --> 00:21:38,920 Maybe height of a person. 320 00:21:38,920 --> 00:21:41,060 Height of a person. 321 00:21:41,060 --> 00:21:47,430 OK, so this is now another example, just to get practice 322 00:21:47,430 --> 00:21:51,940 in the relation between the height of a person and the 323 00:21:51,940 --> 00:21:54,470 rate of change of the height. 324 00:21:54,470 --> 00:21:57,120 So this is the height. 325 00:21:57,120 --> 00:21:58,990 Maybe I'll call it y. 326 00:21:58,990 --> 00:22:04,040 Let me write height of a person. 327 00:22:04,040 --> 00:22:06,840 And what is this going to be? 328 00:22:06,840 --> 00:22:09,950 What is function two? 329 00:22:09,950 --> 00:22:12,510 Well, slope doesn't seem quite right. 330 00:22:12,510 --> 00:22:18,120 The point about function two is it tells how fast function 331 00:22:18,120 --> 00:22:20,000 one changes. 332 00:22:20,000 --> 00:22:25,720 It's the rate of change of the height. 333 00:22:25,720 --> 00:22:28,610 It's the rate of change. 334 00:22:28,610 --> 00:22:34,890 So let me call it s, and it'll be the rate of change. 335 00:22:34,890 --> 00:22:36,620 Good if I use those words. 336 00:22:43,738 --> 00:22:47,120 Yeah, so I want to think just how we grow, a 337 00:22:47,120 --> 00:22:49,540 typical person growing. 338 00:22:49,540 --> 00:22:56,910 In fact, as I wrote this on the board, I thought of 339 00:22:56,910 --> 00:22:58,060 another pair. 340 00:22:58,060 --> 00:23:01,310 Can I just say it in words, this other pair, and then I'll 341 00:23:01,310 --> 00:23:03,330 come back to this one? 342 00:23:03,330 --> 00:23:05,430 Here's another pair. 343 00:23:05,430 --> 00:23:11,880 This could be money in a bank. 344 00:23:15,000 --> 00:23:16,360 Wealth sounds better. 345 00:23:16,360 --> 00:23:17,610 Let's call it wealth. 346 00:23:19,990 --> 00:23:23,340 That's zippier. 347 00:23:23,340 --> 00:23:25,490 And then what is this one? 348 00:23:25,490 --> 00:23:28,840 If this is your wealth, your total 349 00:23:28,840 --> 00:23:32,890 assets, what's your worth? 350 00:23:32,890 --> 00:23:38,730 This would be the rate of change, how 351 00:23:38,730 --> 00:23:41,510 quickly you're saving. 352 00:23:41,510 --> 00:23:42,970 s could be for saving. 353 00:23:42,970 --> 00:23:50,450 Or if you're down here, s is for spending, right? 354 00:23:50,450 --> 00:23:54,550 If s is positive, that means you're wealth is increasing, 355 00:23:54,550 --> 00:23:55,800 you're saving. 356 00:23:58,010 --> 00:24:01,940 Negative s means you're spending, and your wealth goes 357 00:24:01,940 --> 00:24:07,210 whatever, maybe-- 358 00:24:07,210 --> 00:24:08,760 I hope-- up. 359 00:24:08,760 --> 00:24:12,470 Height is mostly up, right? 360 00:24:12,470 --> 00:24:14,270 So let me come back to height of a person. 361 00:24:17,010 --> 00:24:18,360 Now, where-- 362 00:24:18,360 --> 00:24:22,780 oh, and this is time in years. 363 00:24:22,780 --> 00:24:28,340 This is t in years, and this, too, of course. 364 00:24:34,620 --> 00:24:40,340 Actually, I realize you started at t 365 00:24:40,340 --> 00:24:43,120 equals zero: birth. 366 00:24:43,120 --> 00:24:44,835 You do start at a certain-- 367 00:24:47,790 --> 00:24:49,330 actually, what do I know? 368 00:24:52,640 --> 00:24:53,680 You don't say tall. 369 00:24:53,680 --> 00:24:54,530 You say long. 370 00:24:54,530 --> 00:24:56,930 But then as soon as you can stand up, it's tall, 371 00:24:56,930 --> 00:24:58,180 so let's say tall. 372 00:25:02,230 --> 00:25:03,780 Shall we guessed 20 inches? 373 00:25:03,780 --> 00:25:06,720 If that's way off, I apologize to everybody. 374 00:25:06,720 --> 00:25:11,060 Let me just say 20, 20 inches. 375 00:25:11,060 --> 00:25:13,270 OK, at year zero. 376 00:25:13,270 --> 00:25:17,270 OK, and then presumably you grow. 377 00:25:17,270 --> 00:25:19,680 OK, so you grow a little. 378 00:25:19,680 --> 00:25:20,790 What are we headed for? 379 00:25:20,790 --> 00:25:24,450 About 60, 70 inches or something. 380 00:25:24,450 --> 00:25:25,900 Anyway, you grow. 381 00:25:30,230 --> 00:25:34,230 Let's say that's 10 years old and here is 20 years old. 382 00:25:34,230 --> 00:25:35,780 OK, so you grow. 383 00:25:35,780 --> 00:25:37,700 Maybe you grow faster than that. 384 00:25:37,700 --> 00:25:40,990 Let's say you're a healthy person here. 385 00:25:40,990 --> 00:25:43,970 OK, up you grow. 386 00:25:43,970 --> 00:25:55,200 And then at about maybe age 12 or 13, there's a growth spurt. 387 00:25:55,200 --> 00:25:59,030 And maybe the point is, how do we see that growth spurt on 388 00:25:59,030 --> 00:26:00,210 the two graphs? 389 00:26:00,210 --> 00:26:03,320 Differently, but it's the same growth spurt. 390 00:26:03,320 --> 00:26:07,340 OK, so here your height suddenly jumps up. 391 00:26:07,340 --> 00:26:09,330 Boy, yeah, you catch up with everybody. 392 00:26:12,070 --> 00:26:16,500 And then at about 12 or 13 well, then unfortunately, it 393 00:26:16,500 --> 00:26:22,190 doesn't do that forever, and it kind of levels off here. 394 00:26:22,190 --> 00:26:24,960 It levels off, and actually you don't 395 00:26:24,960 --> 00:26:28,300 grow a whole lot more. 396 00:26:28,300 --> 00:26:34,930 In fact, I think when you get to about-- oh, I don't know. 397 00:26:34,930 --> 00:26:35,380 Whatever. 398 00:26:35,380 --> 00:26:37,200 We won't discuss this point. 399 00:26:37,200 --> 00:26:43,900 I say when you get too old, you probably lose some. 400 00:26:43,900 --> 00:26:47,100 Let's not emphasize that. 401 00:26:47,100 --> 00:26:51,230 OK, so here is the-- 402 00:26:51,230 --> 00:26:55,460 now, what's happening over here? 403 00:26:55,460 --> 00:26:58,950 Well, it's the slope of that graph. 404 00:26:58,950 --> 00:27:01,080 So the slope might be-- 405 00:27:01,080 --> 00:27:07,520 this is time zero, but you're growing right away. 406 00:27:07,520 --> 00:27:12,230 The s graph, the rate of growth graph, 407 00:27:12,230 --> 00:27:13,990 doesn't start at zero. 408 00:27:13,990 --> 00:27:17,280 It starts how fast you're growing, whatever you're 409 00:27:17,280 --> 00:27:21,040 growing, whatever that slope is. 410 00:27:21,040 --> 00:27:24,710 It's fantastic that when we draw graphs of things, the 411 00:27:24,710 --> 00:27:27,870 word "slope" is suddenly the right word. 412 00:27:27,870 --> 00:27:32,370 OK, so you're growing, maybe at a pretty good rate here. 413 00:27:32,370 --> 00:27:36,960 And let me mark out 10 and 20 years. 414 00:27:36,960 --> 00:27:43,500 And OK, you're doing well, you're coming along here, and 415 00:27:43,500 --> 00:27:46,050 then the growth spurt. 416 00:27:46,050 --> 00:27:49,295 OK, so then suddenly, your rate of growth takes off. 417 00:27:52,190 --> 00:27:58,090 But it doesn't stay that way, right? 418 00:27:58,090 --> 00:28:01,070 Your rate of growth levels off, in fact, 419 00:28:01,070 --> 00:28:03,990 levels way off, levels-- 420 00:28:03,990 --> 00:28:09,650 you'll come down to here, and you probably don't grow a lot. 421 00:28:09,650 --> 00:28:11,190 Do you see the two? 422 00:28:11,190 --> 00:28:15,000 This was the growth curve. 423 00:28:15,000 --> 00:28:16,940 This was the fast growth. 424 00:28:16,940 --> 00:28:19,090 But then it stopped. 425 00:28:19,090 --> 00:28:21,160 Up here, it slowed down. 426 00:28:21,160 --> 00:28:23,090 Here, it dropped. 427 00:28:23,090 --> 00:28:33,120 And oh, if we allow for this person who lived too long, 428 00:28:33,120 --> 00:28:36,540 height actually drops. 429 00:28:36,540 --> 00:28:41,640 OK, there is an example in which I don't-- 430 00:28:41,640 --> 00:28:48,550 also I'm sure people have devised approximate formulas 431 00:28:48,550 --> 00:28:53,480 for average growth rates, but you see, I'm not-- 432 00:28:53,480 --> 00:28:57,520 it's the idea of the relation between function one and 433 00:28:57,520 --> 00:29:00,770 function two that I'm emphasizing. 434 00:29:00,770 --> 00:29:11,040 Now, my last example, let me take one more example, one 435 00:29:11,040 --> 00:29:12,900 more example for this first lecture. 436 00:29:16,050 --> 00:29:21,990 So let me take a case in which the speed is-- 437 00:29:21,990 --> 00:29:23,483 so here will be my two-- 438 00:29:29,020 --> 00:29:30,210 let's use speed. 439 00:29:30,210 --> 00:29:34,230 Let's use this as distance. 440 00:29:34,230 --> 00:29:41,840 This is distance again, and graph two, as 441 00:29:41,840 --> 00:29:44,045 always, will be speed. 442 00:29:49,670 --> 00:29:53,720 And I'm going to take a case in which 443 00:29:53,720 --> 00:29:55,630 it's given by a formula. 444 00:29:55,630 --> 00:30:02,030 I'm going to let the speed be increasing steadily. 445 00:30:02,030 --> 00:30:07,480 OK, so my speed graph this time is going to go up at a 446 00:30:07,480 --> 00:30:09,810 constant rate. 447 00:30:09,810 --> 00:30:12,690 So this is the speed s. 448 00:30:12,690 --> 00:30:14,780 This is the time t. 449 00:30:14,780 --> 00:30:18,530 So this would be s equals-- 450 00:30:18,530 --> 00:30:20,070 s is proportional to t. 451 00:30:20,070 --> 00:30:22,180 That's where you get a straight line. 452 00:30:22,180 --> 00:30:26,460 s is let's say a times t. 453 00:30:26,460 --> 00:30:30,690 That a, a physicist, if we were physicists, would say 454 00:30:30,690 --> 00:30:32,570 acceleration. 455 00:30:32,570 --> 00:30:34,260 You're accelerating. 456 00:30:34,260 --> 00:30:40,560 You're keeping your foot on the gas, steadily speeding up, 457 00:30:40,560 --> 00:30:47,260 and so then s is proportional to t. 458 00:30:47,260 --> 00:30:54,580 Now, think about the distance. 459 00:30:54,580 --> 00:30:56,600 What's happening with distance? 460 00:30:56,600 --> 00:31:00,540 If this is accelerating, you're 461 00:31:00,540 --> 00:31:01,780 going faster and faster. 462 00:31:01,780 --> 00:31:05,210 You're covering more and more speed, more and more distance, 463 00:31:05,210 --> 00:31:08,420 more and more quickly. 464 00:31:08,420 --> 00:31:11,770 If this is slope, the slope is increasing. 465 00:31:11,770 --> 00:31:12,870 Look, the graph-- 466 00:31:12,870 --> 00:31:15,820 let's start the trip meter at zero. 467 00:31:15,820 --> 00:31:17,920 So you started with a speed of zero. 468 00:31:17,920 --> 00:31:24,120 You were not really increasing distance until you got 469 00:31:24,120 --> 00:31:25,820 slightly beyond zero, and then it 470 00:31:25,820 --> 00:31:27,920 slightly started to increase. 471 00:31:27,920 --> 00:31:36,225 But then it increases faster and faster, right? 472 00:31:36,225 --> 00:31:47,730 It never gets infinitely fast, but it keeps going upwards. 473 00:31:47,730 --> 00:31:50,780 And the calculus question would be 474 00:31:50,780 --> 00:31:52,550 can we give a formula-- 475 00:31:52,550 --> 00:31:53,810 an equation-- 476 00:31:53,810 --> 00:31:55,060 for the distance? 477 00:31:57,820 --> 00:32:01,970 Because in this case, I guess I started with function two, 478 00:32:01,970 --> 00:32:05,710 and therefore, it's function one that I want to look at. 479 00:32:05,710 --> 00:32:08,550 It's always pairs of functions. 480 00:32:08,550 --> 00:32:17,260 OK, now, let's think where this would actually happen. 481 00:32:17,260 --> 00:32:22,040 If we were leaning over the Tower of Pisa or whatever, 482 00:32:22,040 --> 00:32:25,220 like Galileo, and drop something, or even just drop 483 00:32:25,220 --> 00:32:31,700 something anywhere, that would be-- we drop it. 484 00:32:31,700 --> 00:32:35,310 At the beginning, it has no speed, but of course, 485 00:32:35,310 --> 00:32:39,310 instantly it picks up speed. 486 00:32:39,310 --> 00:32:42,580 The a would have something to do with the gravitational 487 00:32:42,580 --> 00:32:45,820 constant for the Earth, whatever, and then maybe-- 488 00:32:45,820 --> 00:32:47,880 yeah. 489 00:32:47,880 --> 00:32:50,580 And what would be the distance? 490 00:32:50,580 --> 00:32:56,810 OK, now can I just mention a small miracle of calculus? 491 00:32:56,810 --> 00:32:59,810 A small miracle. 492 00:32:59,810 --> 00:33:03,270 I'm going this direction now from speed to distance so I'm 493 00:33:03,270 --> 00:33:05,600 doing integral calculus. 494 00:33:05,600 --> 00:33:11,730 And we'll get to that later. 495 00:33:11,730 --> 00:33:13,500 In the first lectures, we're almost always 496 00:33:13,500 --> 00:33:15,320 going from one to two. 497 00:33:15,320 --> 00:33:20,190 But here is a neat fact about going from two to one, that if 498 00:33:20,190 --> 00:33:26,580 this is the time t, then, of course, this height here will 499 00:33:26,580 --> 00:33:31,600 be a times t. 500 00:33:31,600 --> 00:33:40,710 And the amazing fact is that this graph tells you the area 501 00:33:40,710 --> 00:33:42,830 under this one. 502 00:33:42,830 --> 00:33:44,550 Graph one-- 503 00:33:44,550 --> 00:33:45,630 function one-- 504 00:33:45,630 --> 00:33:50,970 tells you the area under the graph two. 505 00:33:50,970 --> 00:33:56,640 And in this example with a nice constant acceleration, 506 00:33:56,640 --> 00:34:00,550 steady increase in speed, we know this. 507 00:34:00,550 --> 00:34:02,000 This is a triangle. 508 00:34:02,000 --> 00:34:03,890 It has a base of t. 509 00:34:03,890 --> 00:34:06,470 It has a height of at. 510 00:34:06,470 --> 00:34:12,610 And the area of a triangle, of course, the area, and my point 511 00:34:12,610 --> 00:34:16,050 is that the area is function one-- 512 00:34:16,050 --> 00:34:22,130 amazing; that's just terrific-- 513 00:34:22,130 --> 00:34:29,090 will be-- the area of this triangle here is 1/2 of the 514 00:34:29,090 --> 00:34:31,560 base times the height. 515 00:34:36,469 --> 00:34:39,310 That's the area, and calculus will tell us 516 00:34:39,310 --> 00:34:41,360 that's function one. 517 00:34:41,360 --> 00:34:50,860 So this function is 1/2 of a times t squared. 518 00:34:50,860 --> 00:34:59,180 So there is a function one, and here is df dt. 519 00:34:59,180 --> 00:35:03,470 If I go back to the first letters that I mentioned, if 520 00:35:03,470 --> 00:35:09,250 this is my function f, then this is my function that-- 521 00:35:09,250 --> 00:35:11,650 and notice what kind of a curve that is. 522 00:35:11,650 --> 00:35:14,250 Do you recognize that with a square? 523 00:35:14,250 --> 00:35:19,060 That tells me it's a parabola, a famous and important curve. 524 00:35:19,060 --> 00:35:20,540 And, of course, it's important because it 525 00:35:20,540 --> 00:35:22,730 has such a neat formula. 526 00:35:22,730 --> 00:35:32,680 OK, so we have found the function one. 527 00:35:32,680 --> 00:35:36,710 We've recovered the information in that lost black 528 00:35:36,710 --> 00:35:43,260 box, the distance box, the trip meter, from what we did 529 00:35:43,260 --> 00:35:50,110 find in black box two, the speed, the record of speed. 530 00:35:50,110 --> 00:35:52,710 And notice, I'm using the speed all the way 531 00:35:52,710 --> 00:35:54,560 from here to here. 532 00:35:54,560 --> 00:35:58,770 The speed kind of tells me how the distance is piled up. 533 00:35:58,770 --> 00:36:03,700 The distance is kind of a running total, where the speed 534 00:36:03,700 --> 00:36:07,610 at that moment is an instant thing. 535 00:36:07,610 --> 00:36:11,060 Oh, we have to do that in future lectures. 536 00:36:11,060 --> 00:36:16,480 The difference between a running total of total 537 00:36:16,480 --> 00:36:21,780 distance covered and a speed that's telling me at a moment, 538 00:36:21,780 --> 00:36:25,120 at an instant how distance is changing. 539 00:36:25,120 --> 00:36:31,050 The slope at this very point t, that slope is 540 00:36:31,050 --> 00:36:33,440 this height, is at. 541 00:36:33,440 --> 00:36:36,160 OK, so there you have the first-- 542 00:36:36,160 --> 00:36:37,850 well, I'll say the second. 543 00:36:37,850 --> 00:36:42,050 The first pair of calculus was this one. 544 00:36:42,050 --> 00:36:46,500 f equals st, and it's derivative was s. 545 00:36:46,500 --> 00:36:50,220 Our second pair is f is this, and will you allow 546 00:36:50,220 --> 00:36:52,210 me to write df dt? 547 00:36:55,100 --> 00:37:05,134 If f is 1/2 of at squared, then df dt is at. 548 00:37:08,250 --> 00:37:10,250 You'll see this rule again. 549 00:37:10,250 --> 00:37:15,680 The power two dropped to a power one. 550 00:37:15,680 --> 00:37:20,790 But the two multiplied the thing so it canceled the 1/2 551 00:37:20,790 --> 00:37:24,390 and just left the a. 552 00:37:24,390 --> 00:37:27,580 OK, that's a start on the highlights of calculus. 553 00:37:27,580 --> 00:37:28,810 Thanks. 554 00:37:28,810 --> 00:37:30,610 NARRATOR: This has been a production of MIT 555 00:37:30,610 --> 00:37:33,000 OpenCourseWare and Gilbert Strang. 556 00:37:33,000 --> 00:37:35,270 Funding for this video was provided by the Lord 557 00:37:35,270 --> 00:37:36,490 Foundation. 558 00:37:36,490 --> 00:37:39,620 To help OCW continue to provide free and open access 559 00:37:39,620 --> 00:37:42,690 to MIT courses, please make a donation at 560 00:37:42,690 --> 00:37:44,250 ocw.mit.edu/donate.